课程名称︰分析导论优一
课程性质︰数学系大二必修
课程教师︰王振男
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2014/09/30
考试时限（分钟）：40
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
1. (10%) A real number is called algebraic if it is a root of an algebraic
n
equation f(x) = 0, where f(x) = a + a x + ... + a x is a polynomial with
0 1 n
integer coefficients. Prove that the set of all polynomials with integer
coefficients is countable and deduce that the set of algebraic numbers is
also countable.
2. (10%) (a) Find the fallacy in the following "proof" of the statement "The
set of all intervals of positive length is countable".
Let {x , x , ...} denote the countable set of rational numbers and let I be
1 2
any interval of positive length. Then I contains infinitely many rational
points x , but among these there will be one with smallest index n. Define
n
a function F by means of the equation F(I) = n, if x is the rational
n
number with smallest index in the interval I. This function establishes a
one-to-one correspondence between the set of all intervals and a subset of
the positive integers. Hence the set of all intervals is countable.
(b) Can you revise the statement so that the proof given above proves the
statement you revise?